SCIENCE CHINA Information Sciences, Volume 63 , Issue 12 : 222303(2020) https://doi.org/10.1007/s11432-020-3033-3

## Statistical CSI based design for intelligent reflecting surface assisted MISO systems

• AcceptedAug 12, 2020
• PublishedOct 30, 2020
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### Acknowledgment

This work was supported by National Key RD Program of China (Grant No. 2019YFB1803400), the NSFC-Zhejiang Joint Fund for the Integration of Industrialization and Informatization (Grant No. U1709219), and National Natural Science Foundation of China (Grant No. 61922071).

### Supplement

Appendix

Proof of Proposition 3.1

Applying the Jensen's inequality, we have \begin{align}{\rm E}\left\{ \log_2 \left( 1+\gamma \right) \right\} \lessapprox \log_2 \left(1+{\rm E}\{\gamma\} \right) = \log_2 \left( 1+\gamma_0 {\rm E} \left\{ \left|({\boldsymbol h}_2^{\rm T} \boldsymbol{\Phi} {\boldsymbol H}_1+\lambda{\boldsymbol g}^{\rm T}) {\boldsymbol f} \right|^2 \right\} \right). \tag{29} \end{align} The remain task is the derivation of ${\rm~E}~\{~\left|({\boldsymbol~h}_2~\boldsymbol{\Phi}~{\boldsymbol~H}_1+\lambda{\boldsymbol~g})~{\boldsymbol~f}~\right|^2~\}$. Applying the binomial expansion theorem, we have \begin{align}& \left| \left({\boldsymbol h}_2^{\rm T} \boldsymbol{\Phi} {\boldsymbol H}_1+\lambda {\boldsymbol g}^{\rm T} \right) {\boldsymbol f} \right|^2 \tag{30} \\ &= \left| \left(a_2 {\bar{\boldsymbol h}}_2^{\rm T}+ b_2 {\tilde{\boldsymbol h}}_2^{\rm T} \right) \boldsymbol{\Phi} \left(a_1 {\bar{\boldsymbol H}}_1 + b_1 {\tilde{\boldsymbol H}}_1 \right) {\boldsymbol f} + \lambda\left(a_0 {\bar{\boldsymbol g}}^{\rm T} + b_0 {\tilde{\boldsymbol g}}^{\rm T} \right) {\boldsymbol f} \right|^2 \\ &= \left| x_1 + x_2 + x_3 + x_4 + x_5 \right|^2, \end{align} where \begin{align}& x_1 = (a_2 a_1 {\bar{\boldsymbol h}}_2^{\rm T} \boldsymbol{\Phi} {\bar{\boldsymbol H}}_1+ \lambda a_0 {\bar{\boldsymbol g}}^{\rm T}) {\boldsymbol f}, \tag{31} \\ & x_2 = a_2 b_1 {\bar{\boldsymbol h}}_2^{\rm T} \boldsymbol{\Phi} {\tilde{\boldsymbol H}}_1 {\boldsymbol f}, \tag{32} \\ & x_3 = b_2 a_1 {\tilde{\boldsymbol h}}_2^{\rm T} \boldsymbol{\Phi} {\bar{\boldsymbol H}}_1 {\boldsymbol f}, \tag{33} \\ & x_4 = b_2 b_1 {\tilde{\boldsymbol h}}_2^{\rm T} \boldsymbol{\Phi} {\tilde{\boldsymbol H}}_1 {\boldsymbol f}, \tag{34} \\ & x_5 = \lambda b_0 {\tilde{\boldsymbol g}}^{\rm T} {\boldsymbol f}. \tag{35} \end{align} It is easy to observe that $x_1$ is a constant and ${\rm~E}\{x_i\}~=~0$ holds for $i~=~2,3,4,5$. Besides, since ${\tilde{\boldsymbol~h}}_2$, ${\tilde{\boldsymbol~H}}_1$ and ${\tilde{\boldsymbol~g}}$ have zero means and are independent with each other, we can derive that \begin{align}& {\rm E} \left\{ \left|({\boldsymbol h}_2^{\rm T} \boldsymbol{\Phi} {\boldsymbol H}_1+\lambda {\boldsymbol g}^{\rm T}) {\boldsymbol f} \right|^2 \right\} \tag{36} \\ & = {\rm E} \{ | x_1 + x_2 + x_3 + x_4 + x_5 |^2 \} \\ & = |x_1|^2 + {\rm E}\{|x_2|^2\} + {\rm E}\{|x_3|^2\} + {\rm E}\{|x_4|^2\} + {\rm E}\{|x_5|^2\}. \end{align} Let ${\boldsymbol~w}~\triangleq\tilde{\boldsymbol~H}_1~{\boldsymbol~f}$. ${\rm~E}\{|x_2|^2\}~\in~\mathbb{C}^{N~\times~1}$ can be expressed as \begin{align}{\rm E}\{|x_2|^2\}= a_2^2 b_1^2 {\rm E} \left\{ \mathsf{tr} \left( \bar{\boldsymbol h}_{2}^{*} \bar{\boldsymbol h}_{2}^{\rm T} {\boldsymbol w} {\boldsymbol w}^{\rm H} \right) \right\}=a_2^2 b_1^2 \mathsf{tr} \left( \bar{\boldsymbol h}_{2}^{*} \bar{\boldsymbol h}_{2}^{\rm T} {\rm E} \left\{ {\boldsymbol w} {\boldsymbol w}^{\rm H} \right\} \right). \tag{37} \end{align} Noticing that ${\rm~E} \left\{~{\boldsymbol~w}~{\boldsymbol~w}^{\rm~H}~\right\}=~\bf{I}_N$, we have \begin{align}{\rm E}\{|x_2|^2\}=a_2^2 b_1^2\mathsf{tr} \left( \bar{\boldsymbol h}_{2}^{*} \bar{\boldsymbol h}_{2}^{\rm T} \bf{I}_N \right)=a_2^2 b_1^2 N. \tag{38} \end{align} Following the similar lines, it can be derived that \begin{align}&{\rm E}\{|x_3|^2\} = b_2^2 a_1^2 \left\|{\bar{\boldsymbol H}_\text{1} {\boldsymbol f}}\right\|^2, \tag{39} \\ &{\rm E}\{|x_4|^2\} = b_2^2 b_1^2 N, \tag{40} \\ &{\rm E}\{|x_5|^2\} = \lambda^2 b_0^2. \tag{41} \end{align} Summing over all the values yields the desired result.

### References

[1] Wu Q, Zhang R. Towards Smart and Reconfigurable Environment: Intelligent Reflecting Surface Aided Wireless Network. IEEE Commun Mag, 2020, 58: 106-112 CrossRef Google Scholar

[2] Cui T J, Qi M Q, Wan X. Coding metamaterials, digital metamaterials and programmable metamaterials. Light Sci Appl, 2014, 3: e218-e218 CrossRef ADS arXiv Google Scholar

[3] Tao Q, Wang J W, and Zhong, C J. Performance analysis of intelligent reflecting surface aided communication systems. IEEE Communications Letters, 2020. Google Scholar

[4] Hu X, Zhong C, Zhu Y. Programmable Metasurface-Based Multicast Systems: Design and Analysis. IEEE J Sel Areas Commun, 2020, 38: 1763-1776 CrossRef Google Scholar

[5] Gao J, Zhong C, Chen X. Unsupervised Learning for Passive Beamforming. IEEE Commun Lett, 2020, 24: 1052-1056 CrossRef Google Scholar

[6] Wu Q, Zhang R. Intelligent Reflecting Surface Enhanced Wireless Network via Joint Active and Passive Beamforming. IEEE Trans Wireless Commun, 2019, 18: 5394-5409 CrossRef Google Scholar

[7] Wu Q, Zhang R. Beamforming Optimization for Wireless Network Aided by Intelligent Reflecting Surface With Discrete Phase Shifts. IEEE Trans Commun, 2020, 68: 1838-1851 CrossRef Google Scholar

[8] Yang Y, Zhang S, Zhang R. IRS-Enhanced OFDMA: Joint Resource Allocation and Passive Beamforming Optimization. IEEE Wireless Commun Lett, 2020, 9: 760-764 CrossRef Google Scholar

[9] You X, Zhang C, Tan X. AI for 5G: research directions and paradigms. Sci China Inf Sci, 2019, 62: 21301 CrossRef Google Scholar

[10] Ding Z, Vincent Poor H. A Simple Design of IRS-NOMA Transmission. IEEE Commun Lett, 2020, 24: 1119-1123 CrossRef Google Scholar

[11] Qi Q, Chen X, Zhong C. Physical layer security for massive access in cellular Internet of Things. Sci China Inf Sci, 2020, 63: 121301 CrossRef Google Scholar

[12] Zhang Y, Zhong C, Zhang Z. Sum Rate Optimization for Two Way Communications With Intelligent Reflecting Surface. IEEE Commun Lett, 2020, 24: 1090-1094 CrossRef Google Scholar

[13] Wang P L, Fang J, and Li H B. Joint beamforming for intelligent reflecting surface-assisted millimeter wave communications. 2019,. arXiv Google Scholar

[14] Zhang J Z, Zhang Y, Zhong C J, and Zhang Z Y. Robust design for intelligent reflecting surfaces assisted MISO systsems. IEEE Communications Letters, 2020. Google Scholar

[15] Taha A, Alrabeiah M, Alkhateeb A. Enabling large intelligent surfaces with compressive sensing and deep learning. 2019,. arXiv Google Scholar

[16] Yang Y, Zheng B, Zhang S. Intelligent Reflecting Surface Meets OFDM: Protocol Design and Rate Maximization. IEEE Trans Commun, 2020, 68: 4522-4535 CrossRef Google Scholar

[17] Zheng B, Zhang R. Intelligent Reflecting Surface-Enhanced OFDM: Channel Estimation and Reflection Optimization. IEEE Wireless Commun Lett, 2020, 9: 518-522 CrossRef Google Scholar

[18] Mishra D, Johansson H. Channel estimation and low-complexity beamforming design for passive intelligent surface assisted MISO wireless energy transfer. In: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2019. 4659--4663. Google Scholar

[19] Han Y, Tang W, Jin S. Large Intelligent Surface-Assisted Wireless Communication Exploiting Statistical CSI. IEEE Trans Veh Technol, 2019, 68: 8238-8242 CrossRef Google Scholar

• Figure 1

(Color online) System model.

•

Algorithm 1 Alternating optimization algorithm

The fractional increase of the objective value is below a threshold $\varepsilon~>~0$.

$\mathbf{Output:}$ ${\boldsymbol~\phi}^{\star}~={\boldsymbol~\phi}_{i}$

and ${\boldsymbol~f}^{\star}={\boldsymbol~f}_{i}$.

$\mathbf{Initialization:}$ Given feasible initial solutions ${\boldsymbol~\phi}_{0}$, ${\boldsymbol~f}_{0}$ and the iteration index $i=0$.

repeat

For given transmit beam ${\boldsymbol~f}_{i}$, calculate the optimal phase shift beam accoring to (14), which yields ${\boldsymbol~\phi}_{i+1}$.

For given phase shift beam ${\boldsymbol~\phi}_{i+1}$, compute the optimal transmit beam according to (17), which yields ${\boldsymbol~f}_{i+1}$.

$i~\leftarrow~i+1$.

until

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